A novel framework for fluid/structure interaction in subject-specific surgical simulations involving elastic cardiac geometries

计算流体动力学最重要的应用之一是血流模拟。然而，实际困难限制了模拟的类型和适用性，从而阻碍了血流数值模拟充分发挥其潜力。巨大的计算费用、由于复杂的几何形状而导致的精度降低以及解决方案缺乏规律性限制了血流模拟的规模和范围。目前超出现有方法范围的一项令人兴奋的应用是模拟旨在修复患病和故障心脏瓣膜的手术。瓣膜几何形状和附近流动模式的复杂性使可靠和预测数值模型的开发变得相当复杂。提供患者特定预后的能力需要一种能够准确解决这些流程的算法。然而，心脏几何形状非常复杂，必须用体积和膜成分来表示，由于患者的瓣膜疾病，这两种成分也可能表现出复杂的不规则性。固/流耦合算法必须具有足够的几何灵活性来解决这些特征并适应虚拟手术引起的变化。最终，为了提供有意义的结果，固体/流体算法必须在不牺牲适应性的情况下提供一定程度的准确性和稳定性。现有的流固相互作用方法无法保证这种水平的功能。一般情况下，几何灵活性会换取更高的订单精度。此外，实际需求需要具有最小时间步长限制的稳定算法，因为准确预测术后行为的愿望伴随着在较长时间间隔内运行模拟的内在需求。提供有效模拟瓣膜手术所需的功能具有挑战性，需要同时解决所有这些问题，而现有方法无法做到这一点。该研究的主要贡献将是开发和应用一种易于处理的二阶数值方法，该方法能够将粘性不可压缩流体与以拉格朗日网格表示的薄且体积几何复杂的弹性固体耦合。流体将通过笛卡尔欧拉网格进行建模，其中嵌入了实体表示，以避免在模拟中的每个时间步重新划分计算域网格的高昂成本。将尽可能使用规则网格。几何灵活性和在整个流体域的任意移动表面上施加各种边界条件的能力是实现既定目标的关键，并将成为开发高阶精确纳维斯托克斯求解器的主要指南。对健康和患病心脏瓣膜附近的血流进行特定于患者的计算流体动力学模拟的好处可能会彻底改变某些病理的治疗。这种功能可以让外科医生设计适合个人的新手术，通过数字预测术后结果来确定是否需要手术，甚至可以用于培训外科住院医生最先进的技术。这项工作将集中于开发一种数值方法，用于在具有挑战性的矫正瓣膜手术病例中检查通过此类手术改变的组织的血流。具体来说，我们的目标是改进法洛四联症和二尖瓣修复的治疗。患有法洛四联症的患者需要人工置换瓣膜，其寿命本身有限，准确确定更换这些瓣膜的时间以纠正肺动脉瓣关闭不全是生死攸关的问题。对于二尖瓣修复等手术，困难的选择在于准确确定哪种类型的矫正最适合特定个体。通过这项工作的成功应用，可以改善何时做出这些关键决策以及许多相关其他决策的确定。

One of the most important applications of computational fluid dynamics has been simulation of blood flow. However, practical difficulties have limited the types and applicability of simulations performed, thus preventing numerical modeling of blood flow from reaching its full potential. Extreme computational expense, reduced order of accuracy due to complex geometry and lack of regularity in solutions have restricted the scale and scope of blood flow simulations. One exciting application currently outside the scope of existing methods is the simulation of surgeries designed to repair diseased and malfunctioning heart valves. The complexity of the valvular geometry and flow patterns in their vicinity complicate considerably the development of reliable and predicative numerical models. The ability to deliver patient specific prognoses demands an algorithm that can accurately resolve these flows. However, cardiac geometry is highly complicated and must be represented with both volumetric and membraneous components, either of which might also exhibit intricate irregularities due to the patient's valvular disease. The solid/fluid coupling algorithm must have sufficient geometric flexibility to resolve these features and to adapt to the changes induced by the virtual surgery. Ultimately, to provide meaningful results the solid/fluid algorithm must deliver a certain level of accuracy and stability without sacrificing adaptability. Existing methods for fluid-solid interaction cannot guarantee this level of functionality. The general case sees geometric flexibility traded for higher order accuracy. Also, practical demands create the need for stable algorithms with minimal time step restrictions as the desire to accurately predict postoperative behavior comes with the inherent need to run simulations over longer time intervals. The challenging nature of providing the functionality needed for effectively simulating valvular surgeries requires addressing all these issues simultaneously and existing methods cannot do this. The primary contribution of the proposed research will be the development and application of a tractable second-order numerical method capable of coupling a viscous incompressible fluid with thin and volumetric geometrically complex elastic solids represented with Lagrangian meshes. The fluid will be modeled by a cartesian Eulerian grid in which the solid representations are embedded to avoid the prohibitive cost of re-meshing the computational domain at each time-step in the simulation. Regular grids will be used wherever possible. Geometric flexibility and the ability to impose a variety of boundary conditions on arbitrary moving surfaces throughout the fluid domain are key to accomplishing the stated goals and will be a primary guide in developing the higher-order accurate Navier-Stokes solver.The benefits of patient-specific computational fluid dynamics simulations of blood flow near healthy and diseased heart valves can potentially revolutionize the treatment of certain pathologies. Such functionality could allow the surgeon to design new procedures tailored to the individual, to determine whether or not surgery is needed by numerically predicting postoperative results and could even be used to train surgical residents in state-of-the-art techniques. This effort will focus on the development of a numerical method for examining blood flow through such surgically altered tissues in the challenging case of corrective valvular surgery. Specifically, we target improvements in treatment for Tetralogy of Fallot and mitral valve repair. Patients born with Tetralogy of Fallot require artificial replacement valves with inherently finite lifespan and accurate determination of the time to replace these valves to correct for pulmonary regurgitation is a matter of life and death. With procedures such as mitral valve repair, the difficult choice lies in determining exactly which type of correction best suits a particular individual. The determination of when to make these critical decisions and many related others could potentially be improved with the successful application of this effort.